Evaluation of the Equivalent Mechanical Properties of Lattice Structures Based on the Finite Element Method

Lattice structures have excellent mechanical properties and can be designed by changing the cellular structure. However, the computing scale is extremely large to directly analyze a large-size structure containing a huge number of lattice cells. Evaluating the equivalent mechanical properties instead of the complex geometry of such lattice cells is a feasible way to deal with this problem. This paper aims to propose a series of formulas, including critical structural and material parameters, to fast evaluate the equivalent mechanical properties of lattice structures. A reduced-order model based on the finite element method and beam theory was developed and verified by comparing it with the corresponding full model. This model was then applied to evaluate the equivalent mechanical properties of 25 types of lattice cells. The effects of the material Young’s modulus and Poisson’s ratio, strut diameter, cell size, and cell number on those equivalent mechanical properties were investigated and discussed, and the linear relationship with the material parameters and the non-linear relationship with the structural parameters were found. Finally, a series of analytical-fitting formulas involving the structural and material parameters were obtained, which allows us to fast predict the equivalent mechanical properties of the lattice cells.


Introduction
Lattice structures, emerging with additive manufacturing, have become a new kind of "star structure". The expected structural physical properties can be obtained by changing the cell structures and array mode [1][2][3], which implies that the requirements on material properties can be greatly lowered. Lattice structures have many excellent mechanical and thermal properties, such as light weight, large specific strength, and stiffness, and thermal insulation, which allows the lattice structures to gather more and more attention and have broad application prospects in the industry [4][5][6].
The mechanical performances of lattice structures depend on not only the materials but also the geometry, such as the strut dimensions [7,8], volume fraction [9,10], topology as well as array mode [11][12][13]. Hajnys et al. [14] designed and tested several lattice structures with different cellular structures and array modes and found that the tetragon vertex lattice structure showed excellent mechanical properties. Han et al. [15] prepared Co-Cr alloy porous scaffolds with cubic close-packed (CCP), face center cube (FCC), body center cube (BCC) and spherical hollow cube (SHC) cells by SLM technology and evaluated their compression properties. The results showed that the compressive modulus and strength of FCC, BCC, SHC, and CCP scaffolds were in descending order. In terms of the influence of the lattice volume fraction on the mechanical properties, Hazeli et al. [16] optimized the volume fraction of porous lattice structures (PLS) using finite element analysis to obtain the are in a rod-like shape and usually with a relatively large aspect ratio (ratio of length to diameter), this geometric characteristic allows us to develop a reduced-order model to evaluate the mechanical properties of the lattice structures. A beam-element theory was applied to develop the mechanical reduced-order model, which is then used to evaluate the equivalent mechanical properties of lattice cells.  Figure 1b shows a three-dimensional beam element placed in the natural coordinate system, for which three kinds of deformation behaviors, including axial displacement, torsion, and pure bending, are considered, and the generalized displacement q e and generalized force P e at the end nodes of the element are given as:  (1) where u, v, w, and θ denote the components of displacement and flexural angle, respectively, and P and M represent the components of concentrated force and moment, respectively. Applying the principle of virtual work: Here, D is the stiffness matrix of the material, and B is the geometric matrix, which is given as:  Figure 1b shows a three-dimensional beam element placed in the natural coordinate system, for which three kinds of deformation behaviors, including axial displacement, torsion, and pure bending, are considered, and the generalized displacement q e and generalized force P e at the end nodes of the element are given as: q e = [u 1 v 1 w 1 θ x1 θ y1 θ z1 u 2 v 2 w 2 θ x2 θ y2 θ z2 ] T P e = [P x1 P y1 P z1 M x1 M y1 M z1 P x2 P y2 P z2 M x2 M y2 M z2 ] T (1) where u, v, w, and θ denote the components of displacement and flexural angle, respectively, and P and M represent the components of concentrated force and moment, respectively. Applying the principle of virtual work: where the virtual work of the stress is given as: and of the generalized node force as: we have the local stiffness equations as: Here, D is the stiffness matrix of the material, and B is the geometric matrix, which is given as: for the axial displacement and torsion and for the pure bending in Oxy surface, similar to the bending in Oxz. The left-hand side term of Equation (5) can yield the stiffness matrix of a beam element in the natural coordinate system, in which the axial displacement stiffness matrix is given as: and torsion stiffness matrix as: and the pure bending stiffness matrix in the Oxy surface as: Here, E and G are Young's modulus and shear modulus of the used material, respectively; L is the strut length; A, J, and I z are the cross-sectional area, torsion constant, and second moment of the area with respect to the natural z-axis, respectively, and A = πD 2 /4, J = πD 4 /32, and I z = πD 4 /64 for a circular cross-section. Assembling the aforementioned local stiffness matrices, we have the total local stiffness matrix K * of the beam element in the natural coordinate system, undertaking a transformation of coordinates using a transformational matrix T, and further obtaining the stiffness matrix of the beam element in the global system as: K e = T T K * T (11) where T is a rotation matrix determined by the pose of the beam element. Assembling each element stiffness matrix, we have the stiffness matrix K of the lattice structure and then the global linear equations K·q = P, solving these equations to finally obtain the displacement and flexural angle of each node.

Accuracy Verification
Since the effectiveness of the beam theory depends on the aspect ratio of the struts in lattice cells, the accuracy of the reduced-order model is verified by comparing the computed mechanical responses of a lattice cell with different aspect ratios with those computed by the corresponding full model (a finite element model with solid elements). Here, the mechanical responses to compression and shear loadings on a BCC cell, for instance, with the aspect ratio of 5, 10, and 20 and a fixed dimension of 10 × 10 × 10 mm 3 , as shown in Figure 2, are computed and compared. For the compression loading, a given displacement w = −10 −5 m is applied to the top four nodes, and the displacement freedom of the bottom four nodes along the z-direction is limited; for the shear loading, u = w = 10 −5 m is given to the nodes on the top and the right surfaces, respectively, and the opposite displacement loadings are applied to the opposite corresponding surfaces. Both models use the material with Young's modulus E 0 = 110 GPa and Poisson's ratio µ 0 = 0.34, and the full model is computed by using C3D8R solid elements on a mesh resolution with an averaged grid size of 0.15 D (D denotes the strut diameter of the cells). The code of the reduced-order model is homemade and runs on the MATLAB 2019B software, and the full model for comparison verification is computed by using the ABAQUS 6.14-4 software. The computer is configured as the processor of Intel(R) Core(TM) i5-7200UCPU@2.50GHz 2.71GHz and RAM of 12GB. full model is computed by using C3D8R solid elements on a mesh resolution with an averaged grid size of 0.15 D (D denotes the strut diameter of the cells). The code of the reduced-order model is homemade and runs on the MATLAB 2019B software, and the full model for comparison verification is computed by using the ABAQUS 6.14-4 software The computer is configured as the processor of Intel(R) Core(TM) i5-7200UCPU@2.50GHz 2.71GHz and RAM of 12GB.  Table 1 lists the computed compressing forces and shearing forces of the reduced order model and the full model, as well as the differences between them. Taking the ful model as the benchmark, the reduced-order model shows 5.98% and 4.56% differences in the compressing and the shearing tests with an aspect ratio of about 5, respectively, and the differences are reduced to 1.25% and 1.18% when the aspect ratio is about 10 and further reduced to 0.68% and 0.67% when the aspect ratio is about 20. The results indicate that the reduced-order model can give an excellent evaluation of the mechanical responses of lattice structures with a large aspect ratio.

Equivalent Mechanical Properties
The mechanical response of a lattice structure depends on the cell topology structural parameter, used material, and even the number of arrayed cells. Meanwhile the equivalent mechanical properties usually show as anisotropic due to the asymmetric cell structure even though the used material is isotropic. In this section, we evaluated the equivalent mechanical properties of 25 types of lattice cells given in Figure 3 (their nomenclatures are listed in Appendix A) and investigated the effects of the materia Young's modulus and Poisson's ratio, strut diameter, cell size, and the number of arrayed cells on the equivalent Young's modulus, shear modulus, and Poisson's ratio. According to the structural symmetry, these cells can be divided into two categories: cubic-symmetry structure (including BCC, VC, VBCC, OC, TCC, RC, FPT, ECC, FECC, AFCC, TC, BCCT  and transverse isotropic structure (including BCCF, FCC, FCCZ, FBCC, FFC, VFC, FC2R  TAC, RD, BCCZ, Table 1 lists the computed compressing forces and shearing forces of the reduced-order model and the full model, as well as the differences between them. Taking the full model as the benchmark, the reduced-order model shows 5.98% and 4.56% differences in the compressing and the shearing tests with an aspect ratio of about 5, respectively, and the differences are reduced to 1.25% and 1.18% when the aspect ratio is about 10 and further reduced to 0.68% and 0.67% when the aspect ratio is about 20. The results indicate that the reduced-order model can give an excellent evaluation of the mechanical responses of lattice structures with a large aspect ratio.

Equivalent Mechanical Properties
The mechanical response of a lattice structure depends on the cell topology, structural parameter, used material, and even the number of arrayed cells. Meanwhile, the equivalent mechanical properties usually show as anisotropic due to the asymmetric cell structure even though the used material is isotropic. In this section, we evaluated the equivalent mechanical properties of 25 types of lattice cells given in Figure 3 (their nomenclatures are listed in Appendix A) and investigated the effects of the material Young's modulus and Poisson's ratio, strut diameter, cell size, and the number of arrayed cells on the equivalent Young's modulus, shear modulus, and Poisson's ratio. According to the structural symmetry, these cells can be divided into two categories: cubic-symmetry structure (including BCC, VC, VBCC, OC, TCC, RC, FPT, ECC, FECC, AFCC, TC, BCCT) and transverse  isotropic structure (including BCCF, FCC, FCCZ, FBCC, FFC, VFC, FC2R, TAC, RD, BCCZ,  BCCE, THC, BCCD).

Method of Numerical Experiments
The equivalent Young's modulus, shear modulus, and Poisson's ratio of a lattice structure can be evaluated by carrying out compression and shear tests in three dimensions. Performing the compression test in the i-dimension can yield Young's modulus Ei and Poisson's ratio μij, and the shear test in the i-j plane can yield the shear modulus Gij. Figure 4 shows the schematics of the compression test in the z-dimension and shear test in the x-z plane, respectively, and thus the equivalent Young's modulus Ez, Poisson's ratio μzx and μzy, and shear modulus Gzx can be calculated as follows:

Method of Numerical Experiments
The equivalent Young's modulus, shear modulus, and Poisson's ratio of a lattice structure can be evaluated by carrying out compression and shear tests in three dimensions. Performing the compression test in the i-dimension can yield Young's modulus E i and Poisson's ratio µ ij , and the shear test in the i-j plane can yield the shear modulus G ij . Figure 4 shows the schematics of the compression test in the z-dimension and shear test in the x-z plane, respectively, and thus the equivalent Young's modulus E z , Poisson's ratio µ zx and µ zy , and shear modulus G zx can be calculated as follows: where L x , L y , and L z are the size of the lattice structure in the x-, y-, and z-dimension, respectively, and u, v, and w denote the average displacements in the corresponding dimension, and α and β are the shear deformations. F denotes either the total compressing force or the total shearing force on a loading surface. The displacement loading was applied in the same way introduced in Section 2.2, to the compression and the shear tests, and let both the compression strain and shear strain be smaller than 0.2%. Additionally, Figure 4 only shows a lattice structure with one cell. However, these numerical experiments will be carried out on a lattice structure with much more cells because the equivalent mechanical properties are not always independent of the number of cells. This issue will be examined and discussed in detail in the following section.
where Lx, Ly, and Lz are the size of the lattice structure in the x-, y-, and z-dimension, respectively, and u, v, and w denote the average displacements in the corresponding dimension, and α and β are the shear deformations. F denotes either the total compressing force or the total shearing force on a loading surface. The displacement loading was applied in the same way introduced in Section 2.2, to the compression and the shear tests, and let both the compression strain and shear strain be smaller than 0.2%. Additionally, Figure 4 only shows a lattice structure with one cell. However, these numerical experiments will be carried out on a lattice structure with much more cells because the equivalent mechanical properties are not always independent of the number of cells. This issue will be examined and discussed in detail in the following section.

Effect of the Number of Lattice Cells
Most equivalent mechanical properties, or at least some of their components, depends on the number of lattice cells due to the mass distribution that must be nonhomogeneous, as a consequence, the equivalent mechanical properties can change with the number of cells even though the used material is isotropic and the cells are arrayed periodically. However, the value of the equivalent mechanical properties always tends to be stable with the increase in the number of lattice cells, owing to the mass distribution

Effect of the Number of Lattice Cells
Most equivalent mechanical properties, or at least some of their components, depends on the number of lattice cells due to the mass distribution that must be non-homogeneous, as a consequence, the equivalent mechanical properties can change with the number of cells even though the used material is isotropic and the cells are arrayed periodically. However, the value of the equivalent mechanical properties always tends to be stable with the increase in the number of lattice cells, owing to the mass distribution usually being more homogeneous in a lattice structure with more cells. In this section, the mechanical tests are carried out on the lattice structures with cells from 1 × 1 × 1 to 10 × 10 × 10, where the displacement loading is set to 10 −5 m, the cell size is 10 × 10 × 10 mm 3 , the strut diameter is 1 mm, and Young's modulus of the used material (TC4 for example) is 110 GPa and Poisson's ratio is 0.34.
The equivalent Young's modulus, shear modulus, and Poisson's ratio of the given 25 cells are evaluated from the corresponding lattice structures with different numbers of cells. Figure 5 shows the changes of the equivalent mechanical properties with the size of the lattice structures (only shows the representative ones). Figure 5a shows the tendency of Young's modulus, E z for example, presenting three kinds of characteristics: going up and then becoming stable (including BCC, BCCF, FC2R, TCC, ECC, THC, TC, BCCD, and BCCT), dropping down and then becoming stable (including VC, FCC, FCCZ, VBCC, FBCC, VFC, RD, RC, FPT, BCCZ, BCCE, FECC, and AFCC), and keeping unchanged (cells FFC, OC, and TAC). The equivalent Poisson's ratio also presents those characteristics of Young's modulus, as shown in Figure 5b. For example, µ zx of FCC, FCCZ, VBCC, RD, FPT, BCCZ, AFCC, and THC are positive, and increase and then become stable with the increase in the lattice size, while µ zx of BCC, BCCZ, FBCC, FC2R, TCC, RC, ECC, BCCZ, TC, BCCD, and BCCT drop down and then they tend to be stable. Moreover, µ zx of VC, FFC, VFC, OC, TAC, and FECC are insensitive to the lattice size. Furthermore, the equivalent Poisson's ratio can also be greater than one, zero, and even negative [31,32], such as FC2R (µ zx and µ zy ) and RD (µ yz and µ xz ) are greater than one, VC (µ xy , µ xz , µ yx , µ yz , µ zx , and µ zy ), VFC (µ xz , µ yz , µ zx , and µ zy ), TAC (µ xz , µ yz , µ zx , and µ zy ), and FECC (µ xy , µ xz , µ yx , µ yz , µ zx , and µ zy ) are zero, and FCC (µ xy , and µ yx ), FCCZ (µ xy , and µ yx ), FBCC (µ xy , and µ yx ), FFC (µ xy , and µ yx ), FC2R (µ xy , and µ yx ), and RD (µ xy , and µ yx ) are negative. These indicate that the equivalent Poisson's ratio mostly depends on the structure of the cell itself rather than the Poisson's ratio of the used material, but not for all. This issue will be examined in the next section. The shear modulus, as shown in Figure 5c, also presents those characteristics of Young's modulus, such as G zx , for example, of BCC, BCCF, VBCC, FC2R, TAC, TCC, BCCZ, BCCE, TC, BCCD, and BCCT increase and then become stable with the increase in the lattice size, VC, FCC, FCCZ, FBCC, VFC, RD, RC, FPT, ECC, FECC, AFCC, and THC decrease and then become stable, and FFC and OC remain unchanged. Additionally, the tendencies of the shear modulus in the same cell can be different, such as G zx and G xy in FBCC. The conclusion that the equivalent mechanical properties of lattice structures can vary with the number of cells was also claimed in [33,34], and the specific relationship greatly depends on the cellular structure. However, all of the equivalent mechanical properties are converged with the increase in the lattice size, yielding essentially stable Young's modulus, Poisson's ratio, and a shear modulus at 8 × 8 × 8. The reason is, that the boundary effect becomes weaker and the structure becomes more homogeneous in the macro perspective as the number of cells becomes larger. Thus, the number of cells 8 × 8 × 8 is used to examine the further effects on the mechanical properties in the next.

Effect of Material Properties
To examine the effect of material properties on the equivalent mechanical properties, we sweep Young's modulus E 0 of the used material from 80 to 160 Gpa and Poisson's ratio of µ 0 from 0.2 to 0.8 in the lattice structures with 8 × 8 × 8 cells.
The equivalent Young's modulus, Poisson's ratio, and shear modulus are computed by the reduced-order model, and their responses to the material Young's modulus are shown in Figure 6 (only shows cells 1 to 8, and the characteristics are similar for the rest).
The results indicate that both the equivalent Young's modulus and shear modulus are varied linearly with the material Young's modulus E 0 , and the equivalent Poisson's ratio is almost independent of E 0 . These results are consistent with the beam theory, where the global stiffness equations K·q = P can be derived as (E 0 K 0 )·q = P, where K 0 excludes the material Young's modulus, and as a consequence, the equivalent Young's modulus and shear modulus linearly depend on the material Young's modulus, which thus yields an independent equivalent Poisson's ratio.

Effect of Material Properties
To examine the effect of material properties on the equivalent mechanical properties, we sweep Young's modulus E0 of the used material from 80 to 160 Gpa and Poisson's ratio of μ0 from 0.2 to 0.8 in the lattice structures with 8 × 8 × 8 cells.
The equivalent Young's modulus, Poisson's ratio, and shear modulus are computed by the reduced-order model, and their responses to the material Young's modulus are shown in Figure 6 (only shows cells 1 to 8, and the characteristics are similar for the rest).
The results indicate that both the equivalent Young's modulus and shear modulus are varied linearly with the material Young's modulus E0, and the equivalent Poisson's ratio is almost independent of E0. These results are consistent with the beam theory, where the global stiffness equations K·q = P can be derived as (E0K0)·q = P, where K0 excludes the material Young's modulus, and as a consequence, the equivalent Young's modulus and shear modulus linearly depend on the material Young's modulus, which thus yields an independent equivalent Poisson's ratio. Unlike the material Young's modulus, the material Poisson's ratio µ 0 cannot be separated linearly from the global stiffness matrix due to the material shear modulus included in the torsion stiffness matrix. Consequently, the effect of material Poisson's ratio µ 0 on the equivalent mechanical properties could be complex. We analyzed all the equivalent Young's moduli, shear moduli, and Poisson's ratio but found that those mechanical properties of most of the given 25 cells are insensitive to the material Poisson's ratio except FCC (G xy ), FCCZ (G xy ), FFC (G xy ), TAC (G yz and G zx ), RD (µ xy , and µ yx ), FECC (G xy , G yz , and G zx ), and THC (E x , E y , µ yx , µ yz , µ xy , and µ xz ), as shown in Figure 7. Even though the range of the material Poisson's ratio in the numerical tests was intentionally extended substantially, 0.2 to 0.8, the largest relative variations are not more than 12% (such as FCC (G xy ), FFC (G xy ), and THC (µ yz )). Actually, Poisson's ratio of most engineering materials is smaller than 0.5 and usually around 0.3~0.35 (such as Al, Mg, Ti, and their alloys), in which the relative variations of the equivalent mechanical properties of all the examined lattice structures are less than 1%. The global stiffness matrix implies that the material Poisson's ratio affects the equivalent mechanical properties via the torsion behavior [35,36], however, the torsion load is usually much smaller than tension, compression, and bending loads in the given 25 cells, and as a consequence, the influence of the material Poisson's ratio on the equivalent mechanical properties is very limited. Unlike the material Young's modulus, the material Poisson's ratio μ0 cannot be separated linearly from the global stiffness matrix due to the material shear modulus included in the torsion stiffness matrix. Consequently, the effect of material Poisson's ratio μ0 on the equivalent mechanical properties could be complex. We analyzed all the equivalent Young's moduli, shear moduli, and Poisson's ratio but found that those mechanical properties of most of the given 25 cells are insensitive to the material Poisson's ratio except FCC (Gxy), FCCZ (Gxy), FFC (Gxy), TAC (Gyz and Gzx), RD (μxy, and μyx), FECC (Gxy, Gyz, and Gzx), and THC (Ex, Ey, μyx, μyz, μxy, and μxz), as shown in Figure 7. Even though the range of the material Poisson's ratio in the numerical tests was intentionally extended substantially, 0.2 to 0.8, the largest relative variations are not more than 12% (such as FCC (Gxy), FFC (Gxy), and THC (μyz)). Actually, Poisson's ratio of most engineering materials is smaller than 0.5 and usually around 0.3~0.35 (such as Al, Mg, Ti, and their alloys), in which the relative variations of the equivalent mechanical properties of all the examined lattice structures are less than 1%. The global stiffness matrix implies that the material Poisson's ratio affects the equivalent mechanical properties via the torsion behavior [35,36], however, the torsion load is usually much smaller than tension, compression, and bending loads in the given 25 cells, and as a consequence, the influence of the material Poisson's ratio on the equivalent mechanical properties is very limited.

Effect of Cell Dimensions
In this section, the effects of strut diameter (D) and cell size (a) on the equivalent mechanical properties were examined. In the study of the strut diameter, the equivalent Young's modulus, shear modulus, and Poisson's ratio of the given 25 cells are computed on the lattice structures with 8 × 8 × 8 cells and a cell size of 10 × 10 × 10 mm 3 as well as a swept diameter from 0.5 to 2 mm. The dependence of the equivalent mechanical properties on the strut diameter was summarized and the representative ones of which are shown in Figure 8. The equivalent Young's modulus and shear modulus, as shown in Figure 8a,b, present a roughly exponential rising tendency with the increase in strut diameter. The situation of the equivalent Poisson's ratio, shown in Figure 8c, is rather complex, and it can go up, drop down, and even keep at zero with the increase in the strut diameter. The exact relationship between the equivalent mechanical properties and the strut diameter is complex and depends on the cell structure. However, this relationship can also be analyzed from the global stiffness matrix K, where the axial displacement stiffness matrix K e ∞D 2 , torsion stiffness matrix K t ∞D 4 , and pure bending stiffness matrix K b ∞D 4 . Consequently, the equivalent Young's modulus and shear modulus are the functions of the square to the fourth power of the strut diameter. This relationship was also presented in [29], where the fitted compressive modulus of BCC is proportional to the strut diameter to the power of 3.5788.

Effect of Cell Dimensions
In this section, the effects of strut diameter (D) and cell size (a) on the equivalent mechanical properties were examined. In the study of the strut diameter, the equivalent Young's modulus, shear modulus, and Poisson's ratio of the given 25 cells are computed on the lattice structures with 8 × 8 × 8 cells and a cell size of 10 × 10 × 10 mm 3 as well as a swept diameter from 0.5 to 2 mm. The dependence of the equivalent mechanical properties on the strut diameter was summarized and the representative ones of which are shown in Figure 8. The equivalent Young's modulus and shear modulus, as shown in Figure 8a,b, present a roughly exponential rising tendency with the increase in strut diameter. The situation of the equivalent Poisson's ratio, shown in Figure 8c, is rather complex, and it can go up, drop down, and even keep at zero with the increase in the strut diameter. The exact relationship between the equivalent mechanical properties and the strut diameter is complex and depends on the cell structure. However, this relationship can also be analyzed from the global stiffness matrix K, where the axial displacement stiffness matrix Ke∞D 2 , torsion stiffness matrix Kt∞D 4 , and pure bending stiffness matrix Kb∞D 4 . Consequently, the equivalent Young's modulus and shear modulus are the functions of the square to the fourth power of the strut diameter. This relationship was also presented in [29], where the fitted compressive modulus of BCC is proportional to the strut diameter to the power of 3.5788. In the study of the cell size, the numerical experiments were done on the 8 × 8 × 8 lattices structures with a cell size of 10 × 10 × 10 mm 3 and 20 × 20 × 20 mm 3 , respectively, and where the ratio of the strut diameter to the cell size is fixed, that is, keeping γ = D/a unchanged. We computed all the equivalent Young's moduli, shear moduli, and Poisson's ratio of the given 25 cells, and found the equivalent mechanical properties were unchanged if the cell structure was magnified proportionally. Figure 9 shows the computed total loading forces under a displacement loading of 10 −5 m of the 25 types of lattice structures. The results show that all the total loading forces in both the compression and shear tests on the lattice structures with a larger cell size are just twice as much as those on the lattice structures with a smaller cell size. Consequently, the equivalent Young's modulus, shear modulus, and Poisson's ratio are the same according to Equation (12). These indicate that the equivalent mechanical properties depend on the ratio of the strut diameter to the cell size instead of the cell size. In the study of the cell size, the numerical experiments were done on the 8 × 8 × 8 lattices structures with a cell size of 10 × 10 × 10 mm 3 and 20 × 20 × 20 mm 3 , respectively, and where the ratio of the strut diameter to the cell size is fixed, that is, keeping γ = D/a unchanged. We computed all the equivalent Young's moduli, shear moduli, and Poisson's ratio of the given 25 cells, and found the equivalent mechanical properties were unchanged if the cell structure was magnified proportionally. Figure 9 shows the computed total loading forces under a displacement loading of 10 −5 m of the 25 types of lattice structures. The results show that all the total loading forces in both the compression and shear tests on the lattice structures with a larger cell size are just twice as much as those on the lattice structures with a smaller cell size. Consequently, the equivalent Young's modulus, shear modulus, and Poisson's ratio are the same according to Equation (12). These indicate that the equivalent mechanical properties depend on the ratio of the strut diameter to the cell size instead of the cell size.

Summary of Equivalent Mechanical Properties
Summarizing the above examinations, we have found: that the equivalent Young's modulus and shear modulus linearly depend on the material Young's modulus E0 and are insensitive to the material Poisson's ratio μ0; the equivalent Poisson's ratio is insensitive to the material properties; all the examined equivalent mechanical properties are the function of the ratio of the strut diameter to the cell size γ, and the equivalent Young's

Summary of Equivalent Mechanical Properties
Summarizing the above examinations, we have found: that the equivalent Young's modulus and shear modulus linearly depend on the material Young's modulus E 0 and are insensitive to the material Poisson's ratio µ 0 ; the equivalent Poisson's ratio is insensitive to the material properties; all the examined equivalent mechanical properties are the function of the ratio of the strut diameter to the cell size γ, and the equivalent Young's modulus and shear modulus are the functions of the square to the fourth power of γ. Table 2 summarizes the data and formulas of the equivalent Young's modulus, shear modulus, and Poisson ratio of the given 25 cells. The instances of the listed 25 cells are computed on the corresponding 8 × 8 × 8 lattice structures with a cell size of 10 × 10 × 10 mm 3 and γ = 0.1 and using the material with Young's modulus E 0 = 110 GPa, Poisson's ratio µ 0 = 0.34. The formulas of the equivalent Young's modulus and shear modulus are given in a form of E 0 (bγ c ) according to the aforementioned summaries and that of the equivalent Poisson's ratio is given in the form of a quadratic polynomial. The parameters of those formulas are determined by fitting the data of the equivalent mechanical properties, which are computed using γ from 0.05 to 0.2 and E 0 from 80 to 160 GPa.                         3.426 ), E z = E 0 (18.53 γ 3.588 ) G xy = E 0 0.373γ 2.274 ), G yz = G zx = E 0 0.9711γ 2.058 ) µ xy = µ yx = −4.467γ 2 + 5.661γ − 1.183 µ xz = µ yz = 0.7961γ 2 − 1.553γ + 0.5501 µ zx = µ zy = −7.469γ 2 − 2.167γ + 2.078 MPa, E z = 1.12 GPa G xy = 668 MPa, G yz = G zx = 666 MPa µ xy = µ yx = 0.956, µ xz = µ yz = 0.017, µ zx = µ zy = 0.493 3.964 ), E z = E 0 1.301γ 2.108 ) G xy = E 0 0.6367γ 2.021 ), G yz = G zx = E 0 (0.6481 γ 2.03 ) µ xy = µ yx = −2.913γ 2 − 0.2487γ + 1.009 µ xz = µ yz = 0.8434γ 2 + 0.1373γ − 0.005075 µ zx = µ zy = −0.6905γ 2 + 1.904 × 10 −4 γ + 0.5001     1.

Conclusions
This work evaluated the equivalent Young's modulus, shear modulus, and Poisson's ratio of 25 types of lattice cells using a beam-theory-based reduced-order model, and examined the effects of cell number, cell dimensions, and Young's modulus and Poisson's ratio of the used material on the equivalent mechanical properties. The conclusions of this work are summarized as follows: (1) A reduced-order mechanical model for a rod-like lattice structure was developed, and its accuracy was validated by comparing it with the solid-element full model, showing a relative difference smaller than 1.25% at the aspect ratio larger than 10. Poisson's ratio of the 25 types of lattice cells were systematically given, which would be very useful to expedite the design and optimization procedures of structures containing a huge number of lattice cells.
We have been aware of the fact that the developed reduced-order model regards the strut of lattice cells as an ideal beam element, while lattice structures made by additive manufacturing technologies are often non-ideal and contain certain geometric defects [37], such as non-ideal node shape, variable cross-section, and non-homogeneous microstructures, which could greatly affect the mechanical performances of lattice structures. Additionally, the equivalent mechanical properties were evaluated by homogenization procedures in this paper, while the homogenization procedures are relatively simple. The aforementioned issues are worth more effort to develop a more applicable model in engineering and carry out more complex mechanical tests for lattice structures in the future.